Proof of lower bound of a nonempty set of real numbers

[tronter]

1. Let A be a nonempty set of real numbers which is bounded below. Let -A be the set of numbers -x, where x \in A. Prove that \inf(A) = -\sup(-A).

Intuitively this makes sense if you draw it on a number line. But I am not sure how to formally prove it.

 

[John Creighto]

1. Let A be a nonempty set of real numbers which is bounded below. Let -A be the set of numbers -x, where x \in A. Prove that \inf(A) = -\sup(-A).

Intuitively this makes sense if you draw it on a number line. But I am not sure how to formally prove it.

If -x is in -A then -x<sup(-A)
this implies:
x>-sup(-A)

Therefor -sup(-A) is a lower bound of A.

Now establish it is the least lower bound (probably via contradiction).